Discrete observations of curves are often smoothed by attaching a penalty to the error sum of squares, and the most popular penalty is the integrated squared second derivative of the function that fits the data. But it has been known since the earliest days of smoothing splines that, if the linear differential operator D^2 is replaced by a more general differential operator L that annihilates most of the variation in the observed curves, then the resulting smooth has less bias and greatly reduced mean squared error.
This talk will show how we can use the data to estimate such a linear differential operator for a system of one or more variables. The differential equations estimated in this way represents the dynamics of the processes being estimated. This idea can be used to estimate a forcing function that defines the output of a linear system, and apply this to handwriting data to show that both the static and dynamic aspects of handwriting are well represented by a surprisingly simple second order differential equation.